BESSEL_JN_ZEROS

This function computes the first nt positive roots of J_n(x) for integer order n. The roots are returned in ascending order and exclude the root at x=0 for n>0.

These roots satisfy:

J_n(x_k) = 0, \quad x_k > 0, \quad k=1,2,\dots,nt

The result is returned as a single-row 2D array for Excel range compatibility.

Excel Usage

=BESSEL_JN_ZEROS(n, nt)

n (int, required): Nonnegative integer order of the Bessel function (dimensionless).
nt (int, required): Number of positive roots to return (count).

Returns (list[list]): Single-row array containing the first nt positive roots of J_n.

Example 1: First four roots for order zero

Inputs:

n	nt
0	4

Excel formula:

=BESSEL_JN_ZEROS(0, 4)

Expected output:

Result
2.40483	5.52008	8.65373	11.7915

Example 2: First three roots for order one

Inputs:

n	nt
1	3

Excel formula:

=BESSEL_JN_ZEROS(1, 3)

Expected output:

Result
3.83171	7.01559	10.1735

Example 3: First five roots for order two

Inputs:

n	nt
2	5

Excel formula:

=BESSEL_JN_ZEROS(2, 5)

Expected output:

Result
5.13562	8.41724	11.6198	14.796	17.9598

Example 4: First two roots for order four

Inputs:

n	nt
4	2

Excel formula:

=BESSEL_JN_ZEROS(4, 2)

Expected output:

Result
7.58834	11.0647

Python Code

from scipy.special import jn_zeros as scipy_jn_zeros

def bessel_jn_zeros(n, nt):
    """
    Compute the first positive zeros of the integer-order Bessel function of the first kind.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.jn_zeros.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        n (int): Nonnegative integer order of the Bessel function (dimensionless).
        nt (int): Number of positive roots to return (count).

    Returns:
        list[list]: Single-row array containing the first nt positive roots of J_n.
    """
    try:
        n = int(n)
        nt = int(nt)
        if n < 0:
            return "Error: n must be greater than or equal to 0"
        if nt <= 0:
            return "Error: nt must be greater than 0"
        roots = scipy_jn_zeros(n, nt)
        return [[float(x) for x in roots.tolist()]]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

Nonnegative integer order of the Bessel function (dimensionless).

nt *

Number of positive roots to return (count).